This paper investigates a class of nonlinear parabolic equations governed by the \((p(x), q(x))\) -Laplacian operator with nonlinear source terms. We focus on establishing the well-posedness of solutions, emphasizing both existence and uniqueness. By utilizing Lebesgue and Sobolev spaces with variable exponents, we develop an appropriate functional framework for analysis. The study demonstrates the existence and uniqueness of weak solutions, without imposing sign constraints on the nonlinear terms. We introduce an alternative approach to the \((p(x), q(x))\) -problem by reformulating it as an equivalent fixed-point problem within an appropriate Banach space. Our method relies on the Leray–Schauder topological degree, supported by new technical estimates.
